What to do regarding a neighbor who has not kept up their property?

The first problem you have is that the law provides no remedy for an unkept yard or area--even one that may bring down your property values--other than whatever local/municipal health or building codes he may be violating (i.e the government agencies that have not been helpful are the ones who*could* help if he were violating any local code--if he is not violating local health, building, etc. code, however, their hands are tied). 
The second problem is that the law does not provide a way to guard against what might happen in the future--that is, a tree falling. IF his tree does fall on and damage your house, especially after he has written notice of the hazard, then you could sue him for the damage--but you can't make him do anything about it in advance, only sue him if/when it happens. (If you have not already, send him a written letter, sent some way you can prove delivery, to make sure he has notice of any hazards, like a potentially dangerous tree, so you could hold him liable later.)
The only thing you can do is you can sue him for the cost to repair any damage done by things under his control, like the damage to the fence from his tree. You could also sue hin for the cost to trap or exterminate the animals that come onto your yard, if you are confident that, if necessary, you could prove that came from or were attracted by the condition of his yard. But that's all: sue him for the provable damages or costs you can trace to his property and what he does with it.